We describe coadjoint orbits for three-step free nilpotent Lie groups. It turns out that two-dimensional orbits have the same structure as coadjoint orbits of the Heisenberg group and the Engel group. We consider a time-optimal problem on three-step free nilpotent Lie groups with a set of admissible velocities in the first level of the Lie algebra. The behavior of normal extremal trajectories with initial covectors lying in two-dimensional coadjoint orbits is studied. Under some broad conditions on the set of admissible velocities (in particular, in the sub-Riemannian case) the corresponding extremal controls are periodic, constant, or asymptotically constant.

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三步自由幂零李群的共伴轨道与时间最优控制问题

我们描述了三步自由幂零李群的共伴轨道。事实证明，二维轨道与海森堡群和恩格尔群的共伴轨道具有相同的结构。我们在李代数的第一级中考虑具有一组容许速度的三步自由幂零李群的时间最优问题。研究了初始协向量位于二维 coadjoint 轨道的正态极值轨迹的行为。在一组可容许速度的一些宽泛条件下（特别是在亚黎曼情况下），相应的极值控制是周期性的、恒定的或渐近恒定的。